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Theorem speiv 1790
Description: Inference from existential specialization, using implicit substitition. (Contributed by NM, 19-Aug-1993.)
Hypotheses
Ref Expression
speiv.1  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
speiv.2  |-  ps
Assertion
Ref Expression
speiv  |-  E. x ph
Distinct variable group:    ps, x
Allowed substitution hints:    ph( x, y)    ps( y)

Proof of Theorem speiv
StepHypRef Expression
1 speiv.2 . 2  |-  ps
2 speiv.1 . . . 4  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
32biimprd 156 . . 3  |-  ( x  =  y  ->  ( ps  ->  ph ) )
43spimev 1789 . 2  |-  ( ps 
->  E. x ph )
51, 4ax-mp 7 1  |-  E. x ph
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103   E.wex 1426
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395
This theorem is referenced by: (None)
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